Hands-on Exercise 2b: 2nd Order Spatial Point Patterns Analysis Methods

1 Overview

Spatial Point Pattern Analysis is the evaluation of the pattern or distribution, of a set of points on a surface. The point can be location of:

• Events such as crime, traffic accident and disease onset, or
• Business services (coffee and fastfood outlets) or facilities such as childcare and eldercare.

We’ll use appropriate functions from spatstat to discover spatial point processes of childcare centres in Singapore.

2 The data

Data sets in this exercise are as follows:

• CHILDCARE (Point Feature Data)
  • This dataset provides both location and attribute information of childcare centres across Singapore.
  • Format: GeoJSON
  • Source: data.gov.sg
• MP14_SUBZONE_WEB_PL (Polygon Feature Data)
  • This dataset provide information of URA 2014 Master Plan Planning Subzone boundary data.
  • Format: ESRI Shapefile.
  • Source: data.gov.sg
• CostalOutline (Polygon Feature Data)
  • This dataset shows the national boundary of Singapore.
  • Format: ESRI Shapefile
  • Source: Singapore Land Authority

3 Installing and loading R packages

The code chunk below install and load following packages into R environment:

• sf, a relatively new R package specially designed to import, manage and process vector-based geospatial data in R.
• spatstat, which has a wide range of useful functions for point pattern analysis. In this hands-on exercise, it will be used to perform 1st- and 2nd-order spatial point patterns analysis and derive kernel density estimation (KDE) layer.
• raster which reads, writes, manipulates, analyses and model of gridded spatial data (i.e. raster). In this hands-on exercise, it will be used to convert image output generate by spatstat into raster format.
• maptools which provides a set of tools for manipulating geographic data. In this hands-on exercise, we mainly use it to convert Spatial objects into ppp format of spatstat.
• tmap which provides functions for plotting cartographic quality static point patterns maps or interactive maps by using leaflet API.

pacman::p_load(sf, raster, spatstat, tmap, tidyverse)

4 Spatial data wrangling

4.1 Importing spatial data

We will use st_read() of sf package will be used to import these three geospatial data sets into R.

Childcare data

Since the childcare_sf simple feature data frame is in the WGS84 geodetic CRS, which is not ideal for geospatial analysis, the st_transform() function from the sf package is used to reproject the data to the SVY21 coordinate system during import.

childcare_sf <- st_read("data/child-care-services-geojson.geojson") %>%
  st_transform(crs = 3414)

Reading layer `child-care-services-geojson' from data source 
  `/Users/cham/project/Geospatial-Analytics/chrismanafe/ISSS626-GAA/hands_on_ex/hands_on_ex02/data/child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84

Let’s verify the crs of the data frame to ensure we’re using EPSG 3414.

st_crs(childcare_sf)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Coastal outline data

Import coastal outline data using st_read() function

sg_sf <- st_read(dsn = "data", layer="CostalOutline")

Reading layer `CostalOutline' from data source 
  `/Users/cham/project/Geospatial-Analytics/chrismanafe/ISSS626-GAA/hands_on_ex/hands_on_ex02/data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21

Let’s check coordinate system of this data frame

st_crs(sg_sf)

Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

Coastal outline data frame is using EPSG 9001 instead of 3414 which is suitable for CRS SVY21. Let’s assign correct EPSG code using st_set_crs() then verify the output.

sg_sf = st_set_crs(sg_sf, 3414)

Warning: st_crs<- : replacing crs does not reproject data; use st_transform for
that

st_crs(sg_sf)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Master Plan 2014 data

Let’s load the Master Plan Planning data using st_read() function

mpsz_sf <- st_read(dsn = "data", 
                layer = "MP14_SUBZONE_WEB_PL")

Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/cham/project/Geospatial-Analytics/chrismanafe/ISSS626-GAA/hands_on_ex/hands_on_ex02/data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

Let’s check coordinate system of this data frame

st_crs(mpsz_sf)

Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

mpsz_sf is also using EPSG 9001 instead of 3414 which is suitable for CRS SVY21. Let’s assign correct EPSG code using st_set_crs() then verify the output.

mpsz_sf <- st_set_crs(mpsz_sf,3414)

Warning: st_crs<- : replacing crs does not reproject data; use st_transform for
that

st_crs(mpsz_sf)

Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

4.2 Mapping the geospatial data sets

After checking the referencing system of each geospatial data data frame, it is also useful for us to plot a map to show their spatial patterns.

tm_shape(mpsz_sf) +
  tm_polygons() +
tm_shape(childcare_sf) +
  tm_dots()

Notice that all the geospatial layers are within the same map extend. This shows that their referencing system and coordinate values are referred to similar spatial context. This is very important in any geospatial analysis.

Alternatively, we can also prepare a pin map by using the code chunk below.

tmap_mode('view')

tmap mode set to interactive viewing

tm_shape(childcare_sf)+
  tm_dots()

tmap_mode('plot')

tmap mode set to plotting

In interactive mode, tmap uses the Leaflet for R API. This interactive pin map offers several advantages, including the ability to freely navigate and zoom around the map. We can also query information of each simple feature (i.e. the point) by clicking of them. Additionally, we can change the background of the map layer. Currently, three internet map layers are available: ESRI.WorldGrayCanvas (the default), OpenStreetMap, and ESRI.WorldTopoMap.

Note

Reminder: After using an interactive map, be sure to switch back to plot mode. Each interactive session consumes a connection, and displaying too many interactive maps (ideally no more than 10) in a single RMarkdown document can cause issues when publishing on Netlify.

5 Geospatial data wrangling

5.1 Converting sf data frame into sp’s Spatial* class

Although simple feature data frame is gaining popularity again sp’s Spatial* classes, there are, however, many geospatial analysis packages require the input geospatial data in sp’s Spatial* classes. In this section, we will learn how to convert simple feature data frame to sp’s Spatial* class.

childcare <- as_Spatial(childcare_sf)
mpsz <- as_Spatial(mpsz_sf)
sg <- as_Spatial(sg_sf)

Notice that the geospatial data have been converted into their respective sp’s Spatial* classes now.

Childcare data

childcare

class       : SpatialPointsDataFrame 
features    : 1545 
extent      : 11203.01, 45404.24, 25667.6, 49300.88  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 2
names       :    Name,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Description 
min values  :   kml_1, <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>018989</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>1, MARINA BOULEVARD, #B1 - 01, ONE MARINA BOULEVARD, SINGAPORE 018989</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>THE LITTLE SKOOL-HOUSE INTERNATIONAL PTE. LTD.</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>08F73931F4A691F4</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 
max values  : kml_999,                  <center><table><tr><th colspan='2' align='center'><em>Attributes</em></th></tr><tr bgcolor="#E3E3F3"> <th>ADDRESSBLOCKHOUSENUMBER</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSBUILDINGNAME</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSPOSTALCODE</th> <td>829646</td> </tr><tr bgcolor=""> <th>ADDRESSSTREETNAME</th> <td>200, PONGGOL SEVENTEENTH AVENUE, SINGAPORE 829646</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSTYPE</th> <td></td> </tr><tr bgcolor=""> <th>DESCRIPTION</th> <td>Child Care Services</td> </tr><tr bgcolor="#E3E3F3"> <th>HYPERLINK</th> <td></td> </tr><tr bgcolor=""> <th>LANDXADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor="#E3E3F3"> <th>LANDYADDRESSPOINT</th> <td>0</td> </tr><tr bgcolor=""> <th>NAME</th> <td>RAFFLES KIDZ @ PUNGGOL PTE LTD</td> </tr><tr bgcolor="#E3E3F3"> <th>PHOTOURL</th> <td></td> </tr><tr bgcolor=""> <th>ADDRESSFLOORNUMBER</th> <td></td> </tr><tr bgcolor="#E3E3F3"> <th>INC_CRC</th> <td>379D017BF244B0FA</td> </tr><tr bgcolor=""> <th>FMEL_UPD_D</th> <td>20200826094036</td> </tr><tr bgcolor="#E3E3F3"> <th>ADDRESSUNITNUMBER</th> <td></td> </tr></table></center> 

Coastal outline data

sg

class       : SpatialPolygonsDataFrame 
features    : 60 
extent      : 2663.926, 56047.79, 16357.98, 50244.03  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 4
names       : GDO_GID, MSLINK, MAPID,              COSTAL_NAM 
min values  :       1,      1,     0,             ISLAND LINK 
max values  :      60,     67,     0, SINGAPORE - MAIN ISLAND 

Master plan 2014 data

mpsz

class       : SpatialPolygonsDataFrame 
features    : 323 
extent      : 2667.538, 56396.44, 15748.72, 50256.33  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 15
names       : OBJECTID, SUBZONE_NO, SUBZONE_N, SUBZONE_C, CA_IND, PLN_AREA_N, PLN_AREA_C,       REGION_N, REGION_C,          INC_CRC, FMEL_UPD_D,     X_ADDR,     Y_ADDR,    SHAPE_Leng,    SHAPE_Area 
min values  :        1,          1, ADMIRALTY,    AMSZ01,      N, ANG MO KIO,         AM, CENTRAL REGION,       CR, 00F5E30B5C9B7AD8,      16409,  5092.8949,  19579.069, 871.554887798, 39437.9352703 
max values  :      323,         17,    YUNNAN,    YSSZ09,      Y,     YISHUN,         YS,    WEST REGION,       WR, FFCCF172717C2EAF,      16409, 50424.7923, 49552.7904, 68083.9364708,  69748298.792 

5.2 Converting generic sp format into spatstat’s ppp format

We can use as.ppp() function of spatstat to convert the spatial data into spatstat’s ppp object format.

childcare_ppp <- as.ppp(childcare_sf)

Warning in as.ppp.sf(childcare_sf): only first attribute column is used for
marks

childcare_ppp

Marked planar point pattern: 1545 points
marks are of storage type  'character'
window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units

Let us plot childcare_ppp and examine the difference.

plot(childcare_ppp)

Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character

Warning: Only 10 out of 1545 symbols are shown in the symbol map

We can take a quick look at the summary statistics of the newly created ppp object

summary(childcare_ppp)

Marked planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1545 character character 

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units

Note

Take note of the warning message regarding duplicates. In spatial point pattern analysis, the presence of duplicate points is a significant concern. The statistical methods used in analyzing spatial point patterns typically assume that the processes are simple, meaning that no two points should coincide.

5.3 Handling duplicated points

We can check the duplication in a ppp object by using the code chunk below.

any(duplicated(childcare_ppp))

[1] FALSE

We’ll use the multiplicity() function to count the number of coincident points

multiplicity(childcare_ppp)

   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [112] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [149] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [186] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [223] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [260] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [297] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [334] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [371] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [408] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [445] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [482] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [519] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [556] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [593] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [630] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [667] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [704] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [741] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [778] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [815] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [852] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [889] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [926] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [963] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1000] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1037] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1074] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1111] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1148] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1185] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1222] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1259] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1296] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1333] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1370] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1407] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1444] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1481] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1518] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

We can find out how many locations have more than one point, as shown in the code chunk below.

sum(multiplicity(childcare_ppp) > 1)

[1] 0

There is no duplicate points here, but if we want to visualize the locations of duplicate points, we can plot the childcare data using the code chunk below.

tmap_mode('view')

tmap mode set to interactive viewing

tm_shape(childcare_sf) +
  tm_dots(alpha=0.4, 
          size=0.05)

tmap_mode("plot")

tmap mode set to plotting

There are three ways to address duplicate issue:

1. The simplest approach is to delete the duplicates. However, this will result in the loss of some potentially valuable point events.

2. Another solution is to use jittering, which introduces a small perturbation to the duplicate points so that they no longer occupy the exact same location.

3. The third approach involves making each point “unique” and then attaching the duplicates as marks, or attributes, to the original points. This method requires analytical techniques that account for these marks.

Jittering approach can be implemented as follows

childcare_ppp_jit <- rjitter(childcare_ppp, 
                             retry=TRUE, 
                             nsim=1, 
                             drop=TRUE)

any(duplicated(childcare_ppp_jit))

[1] FALSE

5.4 Creating owin object

When analyzing spatial point patterns, it is good practice to confine the analysis within a specific geographical area, such as the boundary of Singapore. In spatstat, an object called owin is specifically designed to represent such polygonal regions.

The following code demonstrates how to convert a SpatialPolygon object of Singapore into an owin object in spatstat.

sg_owin <- as.owin(sg_sf)

Let’s display the output using plot() and summary() function

plot(sg_owin)

summary(sg_owin)

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

5.5 Combining point events object and owin object

In this last step of geospatial data wrangling, we will extract childcare events that are located within Singapore

childcareSG_ppp = childcare_ppp[sg_owin]

The output object combined both the point and polygon feature in one ppp object class

summary(childcareSG_ppp)

Marked planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1545 character character 

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

Let’s plot the output object

plot(childcareSG_ppp)

Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character

Warning: Only 10 out of 1545 symbols are shown in the symbol map

5.5.1 Extracting study area

Let us extract the target planning areas.

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")

Plot target planning areas

Punggol

par(mfrow=c(2,2))
plot(pg, main = "Punggol")

Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

Tampines

plot(tm, main = "Tampines")

Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

Choa Chu Kang

plot(ck, main = "Choa Chu Kang")

Warning: plotting the first 10 out of 15 attributes; use max.plot = 15 to plot
all

Jurong West

plot(jw, main = "Jurong West")

Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

5.5.2 Creating owin object

We will convert these sf objects into owin objects that is required by spatstat.

pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)

5.5.3 Combining childcare points and the study area

Using the code below, we can extract childcare facilities within specific regions for further analysis.

childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]

Next, the rescale.ppp() function is used to transform the unit of measurement from meters to kilometers.

childcare_pg_ppp.km = rescale.ppp(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale.ppp(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale.ppp(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale.ppp(childcare_jw_ppp, 1000, "km")

The following code is used to plot the four study areas along with the locations of the childcare centers.

par(mfrow=c(2,2))
plot(childcare_pg_ppp.km, main="Punggol")

Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character

Warning: Only 10 out of 61 symbols are shown in the symbol map

plot(childcare_tm_ppp.km, main="Tampines")

Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character

Warning: Only 10 out of 89 symbols are shown in the symbol map

plot(childcare_ck_ppp.km, main="Choa Chu Kang")

Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character

Warning: Only 10 out of 61 symbols are shown in the symbol map

plot(childcare_jw_ppp.km, main="Jurong West")

Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character

Warning: Only 10 out of 88 symbols are shown in the symbol map

6 Analysing spatial point process using G-Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, we will learn how to compute G-function estimation by using Gest() of spatstat package. We will also learn how to perform monte carlo simulation test using envelope() of spatstat package.

Choa Chu Kang

Let us compute G-function estimation using Gest() of spatstat package for Choa Chu Kang area.

G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Choa Chu Kang is random.
• H₁: The distribution of childcare services in Choa Chu Kang is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)

Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.

plot(G_CK.csr)

Given that the observed line is generally within or close to the upper part of the envelope, it suggests a slight tendency towards clustering, but not strong enough to definitively conclude significant clustering without further statistical analysis.

Tampines

Let us compute G-function estimation using Gest() of spatstat package for Tampines area.

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Tampines is random.
• H₁: The distribution of childcare services in Tampines is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform the hypothesis testing then plot the result.

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)

Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.

plot(G_tm.csr)

The observed function is mostly within or near the upper edge of the envelope, indicating a slight tendency towards clustering. However, since it stays mostly within the envelope, it suggests that the observed clustering is not strong enough to significantly deviate from what might be expected under a random distribution.

7 Analysing spatial point process using F-Function

The F function estimates the empty space function F(r) or its hazard rate h(r) from a point pattern within a window of arbitrary shape. In this section, we will learn how to compute the F-function estimation using the Fest() function from the spatstat package. Additionally, we will learn how to perform a Monte Carlo simulation test using the envelope() function from the same package.

Choa Chu Kang

Let us compute F-function estimation using Fest() of spatstat package for Choa Chu Kang area.

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Choa Chu Kang is random.
• H₁: The distribution of childcare services in Choa Chu Kang is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform Monte Carlo test with F-function and plot the test result.

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)

Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.

plot(F_CK.csr)

The F-function plot indicates that the distribution of childcare centers in Choa Chu Kang is generally consistent with a random spatial pattern. There is no strong evidence of significant clustering or dispersion, as the observed pattern falls within the expected range under CSR.

Tampines

Let us compute F-function estimation using Fest() of spatstat package for Tampines area.

F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Tampines is random.
• H₁: The distribution of childcare services in Tampines is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform the hypothesis testing then plot the result.

F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)

Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.

plot(F_tm.csr)

The F-function plot for Tampines suggests that the distribution of childcare centers is more dispersed than what would be expected under complete spatial randomness. The childcare centers are spread out more than a random pattern would predict, indicating a lower density of centers across the area compared to a random distribution.

8 Analysing spatial point process using K-Function

The K-function measures the number of events found up to a given distance from any particular event. In this section, we will learn how to compute K-function estimates using the Kest() function from the spatstat package. We will also learn how to perform a Monte Carlo simulation test using the envelope() function from the same package.

Choa Chu Kang

Let us compute K-function estimation using Kest() of spatstat package for Tampines area.

K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Choa Chu Kang is random.
• H₁: The distribution of childcare services in Choa Chu Kang is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform the hypothesis testing then plot the result.

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)

Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.

plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

The K-function plot for Choa Chu Kang suggests that the distribution of childcare centers is consistent with a random spatial pattern. The observed number of centers within various distances from any given center aligns with what would be expected under complete spatial randomness, indicating no significant clustering or dispersion.

Tampines

Let us compute K-function estimation using Kest() of spatstat package for Tampines area.

K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r, 
     ylab= "K(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Tampines is random.
• H₁: The distribution of childcare services in Tampines is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform the hypothesis testing then plot the result.

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)

Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.

plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

The K-function plot for Tampines reveals significant clustering in the distribution of childcare centers. The observed number of centers within various distances from any given center is much higher than what would be expected under complete spatial randomness, indicating a strong pattern of clustering.

9 Analysing spatial point process using L-Function

The L-function is a transformation of the K-function, providing a more linearized measure of spatial clustering or dispersion. It helps in interpreting the spatial pattern of events by normalizing the K-function and making deviations from complete spatial randomness easier to visualize.

In this section, we will learn how to compute L-function estimation using the Lest() function from the spatstat package. We will also learn how to perform a Monte Carlo simulation test using the envelope() function from the same package.

Choa Chu Kang

Let us compute L-function estimation using Lest() of spatstat package for Choa Chu Kang area.

L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)")

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Choa Chu Kang is random.
• H₁: The distribution of childcare services in Choa Chu Kang is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform the hypothesis testing then plot the result.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)

Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.

plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

The L-function plot for Choa Chu Kang suggests that the distribution of childcare centers is largely consistent with a random spatial pattern. While there are minor fluctuations, the overall pattern does not show significant clustering or dispersion, as the observed L-function generally falls within the expected range under CSR.

Tampines

Let us compute L-function estimation using Lest() of spatstat package for Tampines area.

L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

To confirm the observed spatial patterns, a hypothesis test will be conducted. The hypotheses for the test are as follows:

• H₀: The distribution of childcare services in Tampines is random.
• H₁: The distribution of childcare services in Tampines is not random.

The null hypothesis will be rejected if the p-value is smaller than the alpha value of 0.001.

Let us perform the hypothesis testing then plot the result.

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)

Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.

plot(L_tm.csr, . - r ~ r, 
     xlab="d", ylab="L(d)-r", xlim=c(0,500))

The L-function plot for Tampines reveals significant clustering in the distribution of childcare centers. The observed L-function consistently exceeds the envelope expected under complete spatial randomness, indicating that the centers are more densely packed together than a random distribution would predict. This strong deviation from randomness highlights the presence of spatial clustering in the area.

10 Reference

Kam, T. S. 2nd Order Spatial Point Patterns Analysis Methods. R for Geospatial Data Science and Analytics. https://r4gdsa.netlify.app/chap05